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Chapter 1: Problem 91

The density of mercury changes approximately linearly with temperature as \(\rho_{\mathrm{Hg}}=13595-2.5 \mathrm{~T}\) \(\mathrm{kg} / \mathrm{m}^{3}(T\) in Celsius), so the same pressure difference will result in a manometer reading that is influenced by temperature. If a pressure difference of \(100 \mathrm{kPa}\) is measured in the summer at \(35^{\circ} \mathrm{C}\) and in the winter at \(-15^{\circ} \mathrm{C}\), what is the difference in column height between the two measurements?

Short Answer

Expert verified
The difference in column height is 3 mm.

Step by step solution

01

Understand the Formula for Density

The formula provided for the density of mercury, \( \rho_{\mathrm{Hg}} = 13595 - 2.5 T \), shows that the density decreases linearly with temperature \( T \) in degrees Celsius. This means that as the temperature increases, the density of mercury decreases.
02

Calculate Density at 35掳C

Substitute \( T = 35 \) into the mercury density formula:\[ \rho_{\mathrm{Hg}} = 13595 - 2.5 \times 35 = 13595 - 87.5 = 13507.5 \text{ kg/m}^3 \]
03

Calculate Density at -15掳C

Substitute \( T = -15 \) into the mercury density formula:\[ \rho_{\mathrm{Hg}} = 13595 - 2.5 \times (-15) = 13595 + 37.5 = 13632.5 \text{ kg/m}^3 \]
04

Use the Pressure Difference Formula

The pressure difference \( \Delta P \) that a column of mercury supports is given by the formula \( \Delta P = \rho g h \), where \( g \) is the acceleration due to gravity and \( h \) is the column height.
05

Calculate Column Height at 35掳C

Rearrange the formula \( \Delta P = \rho g h \) to find height \( h \):\[ h = \frac{\Delta P}{\rho g} \]Substitute \( \Delta P = 100,000 \text{ Pa} \), \( \rho = 13507.5 \text{ kg/m}^3 \), and \( g = 9.81 \text{ m/s}^2 \):\[ h_{35} = \frac{100,000}{13507.5 \times 9.81} \approx 0.749 \text{ m} \]
06

Calculate Column Height at -15掳C

Substitute the density at \( -15掳C \) into the height formula:\[ h_{-15} = \frac{100,000}{13632.5 \times 9.81} \approx 0.746 \text{ m} \]
07

Determine the Difference in Column Heights

Calculate the difference between the two column heights:\[ \Delta h = h_{35} - h_{-15} = 0.749 \text{ m} - 0.746 \text{ m} = 0.003 \text{ m} \]
08

Convert the Difference to Millimeters

The difference in meters can be converted to millimeters by multiplying by 1000:\[ \Delta h = 0.003 \text{ m} \times 1000 = 3 \text{ mm} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density of Mercury
Mercury is a heavy liquid metal often used in devices like manometers due to its dense properties. The density of mercury is influenced by temperature, and it changes in a predictable way as described by a linear equation. This equation is given as \[ \rho_{\text{Hg}} = 13595 - 2.5 T \] where \( \rho_{\text{Hg}} \) is the density in \( \text{kg/m}^3 \) and \( T \) is the temperature in Celsius.
  • At higher temperatures, mercury becomes less dense.
  • At lower temperatures, its density increases.
This property is significant when working with devices like manometers, where precise measurements depend on knowing the fluid's exact density. Calculating density accurately becomes crucial because any variation affects pressure readings directly.
Linear Temperature Dependence
The density equation for mercury illustrates a linear temperature dependence. This means a straight-line relationship between the temperature and the density鈥攕hown through the equation\[ \rho_{\text{Hg}} = 13595 - 2.5 T \].Let's simplify what this really means:
  • The density decreases by \( 2.5 \text{ kg/m}^3 \) for every degree Celsius increase.
  • This linear behavior helps predict changes in density just by knowing the temperature change.
The predictability provided by such a linear relationship allows precise adjustments in practical applications, such as scientific experiments or when specific pressures need monitoring.
Pressure Difference
Pressure difference (\( \Delta P \)) is crucial in applications involving manometers, as these devices measure the pressure by the height of a liquid column.The relationship is given by the formula\[ \Delta P = \rho g h \]where \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity (\( 9.81 \text{ m/s}^2 \)), and \( h \) is the height of the column.
  • Increased density results in a greater pressure difference for the same column height.
  • Higher heights generate larger pressure differences, assuming constant density.
When temperature changes, it affects density, and therefore, the pressure calculation also needs to be revisited to ensure accuracy in measurements.
Column Height Measurement
Column height (\( h \)) is a primary measurement in using manometers, used to determine the pressure of a gas or liquid. Changes in the height of the liquid column directly reflect changes in pressure or temperature if one keeps the density constant. The formula to find column height based on a known pressure difference is\[ h = \frac{\Delta P}{\rho g} \].
  • Higher pressure differences lead to increased column heights when other variables remain constant.
  • Lower density requires a taller column to offset the same pressure, due to the volume requirement.
By computing column height adjustments based on changes in density, one can accurately determine pressure differences in varying temperature conditions, such as the ones indicated in the original exercise.

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Most popular questions from this chapter

The standard pressure in the atmosphere with elevation \((H)\) above sea level can be correlated as \(P=\) \(P_{0}(1-H / L)^{5.26}\) with \(L=44300 \mathrm{~m} .\) With the local sea level pressure \(P_{0}\) at \(101 \mathrm{kPa}\), what is the pressure at \(10000 \mathrm{~m}\) elevation?

What is the temperature of \(-5 \mathrm{~F}\) in degrees Rankine?

At the beach, atmospheric pressure is 1025 mbar. You dive \(15 \mathrm{~m}\) down in the ocean, and you later climb a hill up to \(250 \mathrm{~m}\) in elevation. Assume that the density of water is about \(1000 \mathrm{~kg} / \mathrm{m}^{3}\) and the density of air is \(1.18 \mathrm{~kg} / \mathrm{m}^{3}\). What pressure do you feel at each place?

A laboratory room has a vacuum of \(0.1 \mathrm{kPa}\). What net force does that put on the door of size \(2 \mathrm{~m}\) by \(1 \mathrm{~m} ?\)

The difference in height between the columns of a manometer is \(200 \mathrm{~mm}\), with a fluid of density 900 \(\mathrm{kg} / \mathrm{m}^{3}\). What is the pressure difference? What is the height difference if the same pressure difference is measured using mercury (density \(\left.=13600 \mathrm{~kg} / \mathrm{m}^{3}\right)\) as manometer fluid?
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